On the relation between elementary partial difference equations and partial differential equations

Grid functions of nonstandard analysis in the theory of distributions and in partial differential equations

“…If n ∈ * N, D n i is recursively defined as D i (D n−1 i ) and, if α is a multi-index, then D α is defined as expected: For further details about the properties of the finite difference operators, we refer to Hanqiao, St. Mary and Wattenberg [31], to Keisler [33] and to van den Berg [9,10].…”

“…A possibility is to give a nonstandard representation of a given time dependent PDE by discretizing in time as well as in space, and by defining a standard solution to the original problem by the technique of stroboscopy. In [10], van den Berg showed how the stroboscopy technique can be extended to the study of a class of partial differential equations of the first and the second order by imposing additional regularity hypotheses on the time-step of the discretization. For an in-depth discussion on the stroboscopy technique and its applications to partial differential equations, we remand to [10,45,46].…”

“…In [10], van den Berg showed how the stroboscopy technique can be extended to the study of a class of partial differential equations of the first and the second order by imposing additional regularity hypotheses on the time-step of the discretization. For an in-depth discussion on the stroboscopy technique and its applications to partial differential equations, we remand to [10,45,46]. A delicate point in the time discretization of PDEs is that the discrete time step cannot be chosen arbitrarily.…”

Asymptotics of families of solutions of nonlinear difference equations

“…Now (Ŷ (ω + 2) −Ŷ (ω))/Ỹ (ω) = ∅ · (|F 2 (ω,Ỹ (ω))| − 1) becauseŷ ω is of class S 1 under T ω,Ỹ (ω),1/γ ω , respectively M ω,Ỹ (ω) , and by formula (6), and ((Ỹ (ω +2)−Ŷ (ω + 2)) − (Ỹ (ω) −Ŷ (ω)))/Ỹ (ω) = ∅ · (|F 2 (ω,Ỹ (ω))| − 1) becauseỹ ω −ŷ ω is of class |S| 1 under T ω,Ỹ (ω),1/γ ω , respectively M ω,Ỹ (ω) , and by formula (6) or formula (7). The formulae (45) and (46) have obvious geometric interpretations.…”

10.4115/jla.v5i0.173

“…The case for differentiability of first order had already been proved earlier [8] [4]. Then its shadow f is a real function of class C 1 and φ [1] (ξ) f (ξ) for all limited ξ ∈ X.…”